File size: 3,220 Bytes
e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.RingTheory.DirectFiniteness.HoleExistence
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8
noncomputable section
open scoped Classical BigOperators
namespace KaplanskyCounterexample.FiniteGeometry
/-- In a cyclic sequence of at most five positions with unequal neighboring
labels, no label can occur thrice. This includes the wrap-around adjacency. -/
lemma cyclic_label_count_le_two {k : ℕ} {L : Type u_1} [DecidableEq L] (lab : Fin k → L)
(hk : k ≤ 5) (hadj : ∀ i, lab i ≠ lab (finRotate k i)) (l : L) :
((Finset.univ : Finset (Fin k)).filter (fun i => lab i = l)).card ≤ 2 := by
let s := (Finset.univ : Finset (Fin k)).filter (fun i => lab i = l)
have hd : Disjoint s (s.image (finRotate k)) := by
rw [Finset.disjoint_left]
intro i hi hir
obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hir
exact hadj j ((Finset.mem_filter.mp hj).2.trans (Finset.mem_filter.mp hi).2.symm)
have hc : s.card + (s.image (finRotate k)).card ≤ k := by
rw [← Finset.card_union_of_disjoint hd]
exact (Finset.card_le_card (Finset.subset_univ _)).trans_eq (Fintype.card_fin k)
rw [Finset.card_image_of_injective _ (finRotate k).injective] at hc
dsimp [s] at hc
omega
/-- A nonzero vector cannot be the sole contribution to a zero sum. -/
lemma count_ne_one_of_sum_eq_zero {I : Type u_2} {L : Type u_3} {M : Type u_4} [Fintype I] [DecidableEq L]
[AddCommGroup M] (lab : I → L) (v : I → M) (l : L)
(hn : ∀ i, lab i = l → v i ≠ 0)
(hz : ∑ i ∈ (Finset.univ : Finset I).filter (fun i => lab i = l), v i = 0) :
((Finset.univ : Finset I).filter (fun i => lab i = l)).card ≠ 1 := by
intro hc
obtain ⟨i,hi⟩ := Finset.card_eq_one.mp hc
have hi' : i ∈ (Finset.univ : Finset I).filter (fun i => lab i = l) := by rw [hi]; simp
have hil : lab i = l := (Finset.mem_filter.mp hi').2
rw [hi,Finset.sum_singleton] at hz
exact hn i hil hz
lemma cyclic_label_count_eq_two {k : ℕ} {L : Type u_5} {M : Type u_6} [DecidableEq L] [AddCommGroup M]
(lab : Fin k → L) (v : Fin k → M) (hk : k ≤ 5)
(hadj : ∀ i, lab i ≠ lab (finRotate k i)) (l : L)
(hne : ∃ i, lab i = l) (hn : ∀ i, lab i = l → v i ≠ 0)
(hz : ∑ i ∈ (Finset.univ : Finset (Fin k)).filter (fun i => lab i = l), v i = 0) :
((Finset.univ : Finset (Fin k)).filter (fun i => lab i = l)).card = 2 := by
have hle := cyclic_label_count_le_two lab hk hadj l
have hne1 := count_ne_one_of_sum_eq_zero lab v l hn hz
have hp : 0 < ((Finset.univ : Finset (Fin k)).filter (fun i => lab i = l)).card := by
rw [Finset.card_pos]
obtain ⟨i,hi⟩ := hne
exact ⟨i,by simp [hi]⟩
omega
/-- Occurrence-count form of the pigeonhole bound, retaining the exact support. -/
lemma sum_label_counts {I : Type u_7} {L : Type u_8} [Fintype I] [DecidableEq L] (lab : I → L) :
∑ l ∈ Finset.univ.image lab,
((Finset.univ : Finset I).filter (fun i => lab i = l)).card = Fintype.card I := by
symm
simpa using Finset.card_eq_sum_card_fiberwise (s := (Finset.univ : Finset I))
(t := Finset.univ.image lab) (f := lab) (fun i hi => Finset.mem_image_of_mem lab hi)
end KaplanskyCounterexample.FiniteGeometry
end
end OAI
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